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Leave $ A $ be an upper triangular matrix of blocks; The blocks in the diagonal being. $ a in M_p ({ mathbb C}) $ Y $ c in M_q ({ mathbb C}) $.

Leave $ P $ be a polynomial such that $ P (A) = 0_n $.
Leave $ (i, j) $ be two such indices $ j le p <i $.
It seems that $ dP (A) cdot E_ {ij} = 0 $.

Thanks for your interest.